This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional BV spaces. We show that optimality of the regularizing effect for the initial value problem from \(L^\infty \) to fractional Sobolev space and fractional BV spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover, we prove sharp smoothing in \(BV^s\) for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.

本文讨论了守恒律熵解的最佳正则性。为此，我们使用两个关键要素：（a）熵解的精细结构和（b）分数BV空间。我们表明，对于从\（L ^ \ infty \）到分数Sobolev空间和分数BV空间的初值问题，正则化效果的最优性在所有时间内都是有效的。以前，在波的非线性相互作用之前，仅在有限时间内证明了这种最优性。在这里，对于一些精心选择的示例，在波浪相互作用之后获得了鲜明的规律性。而且，我们证明了\（BV ^ s \）中的急剧平滑对于带有线性源项的凸标量守恒定律。接下来，我们为非线性标量多维守恒律和一维或多维的某些双曲系统提供了最大平滑效果的上限。